Wonderful! Thanks for the assistance.

Tom La Bone

-----Original Message-----

From: jiho [mailto:

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Sent: Friday, June 01, 2007 8:17 AM

To:

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Cc: 'R-help'

Subject: Re: [R] Problem with Weighted Variance in Hmisc

On 2007-June-01 , at 13:00 , Tom La Bone wrote:

> The equation for weighted variance given in the NIST DataPlot

> documentation

> is the usual variance equation with the weights inserted. The

> weighted

> variance of the weighted mean is this weighted variance divided by N.

>

> There is another approach to calculating the weighted variance of the

> weighted mean that propagates the uncertainty of each term in the

> weighted

> mean (see Data Reduction and Error Analysis for the Physical

> Sciences by

> Bevington & Robinson). The two approaches do not give the same

> answer. Can

> anyone suggest a reference that discusses the merits of the DataPlot

> approach versus the Bevington approach?

I am no expert but I did a little research on the subject and it

seems there is no analytical equivalent to the standard error of the

mean in weighted statistics (i.e. there is no standard error/variance

of the weighted mean). That's why you find so many formulas: they are

all numerical approximations that make more of less sense. If you

have access to DcienceDirect there is a paper which compares 3 of

those analytical approximations to the variance estimated by bootstrap:

@article{Gatz1995AE,

Author = {Gatz, Donald F and Smith, Luther},

Date-Added = {2007-05-19 14:15:58 +0200},

Date-Modified = {2007-06-01 14:12:17 +0200},

Filed = {Yes},

Journal = {Atmospheric Environment},

Number = {11},

Pages = {1185--1193},

Rating = {0},

Title = {The standard error of a weighted mean concentration--I.

Bootstrapping vs other methods},

Url =

{

http://www.sciencedirect.com/science/article/B6VH3-3YGV47C-6X/
2/18b627259a75ff9b765410aaa231e352},

Volume = {29},

Year = {1995},

Abstract = {Concentrations of chemical constituents of precipitation

are frequently expressed in terms of the precipitation-weighted mean,

which has several desirable properties. Unfortunately, the weighted

mean has no analytical analog of the standard error of the arithmetic

mean for use in characterizing its statistical uncertainty. Several

approximate expressions have been used previously in the literature,

but there is no consensus as to which is best. This paper compares

three methods from the literature with a standard based on

bootstrapping. Comparative calculations were carried out for nine

major ions measured at 222 sampling sites in the National Atmospheric

Deposition/National Trends Network (NADP/NTN). The ratio variance

approximation of Cochran (1977) gave results that were not

statistically different from those of bootstrapping, and is suggested

as the method of choice for routine computing of the standard error

of the weighted mean. The bootstrap method has advantages of its own,

including the fact that it is nonparametric, but requires additional

effort and computation time.}}

The analytical formula which is the closest to the bootstrap is this

one:

var.wtd.mean.cochran <- function(x,w)

#

# Computes the variance of a weighted mean following Cochran 1977

definition

#

{

n = length(w)

xWbar = wtd.mean(x,w)

wbar = mean(w)

out = n/((n-1)*sum(w)^2)*(sum((w*x-wbar*xWbar)^2)-2*xWbar*sum((w-

wbar)*(w*x-wbar*xWbar))+xWbar^2*sum((w-wbar)^2))

return(out)

}

It's the one I am retaining.

NB: the part two of the paper cited above may also interest you:

@article{Gatz1995AEa,

Author = {Gatz, Donald F and Smith, Luther},

Date-Added = {2007-05-19 14:15:58 +0200},

Date-Modified = {2007-06-01 12:11:53 +0200},

Filed = {Yes},

Journal = {Atmospheric Environment},

Number = {11},

Pages = {1195--1200},

Title = {The standard error of a weighted mean concentration--II.

Estimating confidence intervals},

Url =

{

http://www.sciencedirect.com/science/article/B6VH3-3YGV47C-6Y/
2/a187487377ef52b741e3dabdfca97517},

Volume = {29},

Year = {1995},

Abstract = {One motivation for estimating the standard error, SEMw,

of a weighted mean concentration, Mw, of an ion in precipitation is

to use it to compute a confidence interval for Mw. Typically this is

done by multiplying the standard error by a factor that depends on

the degree of confidence one wishes to express, on the assumption

that the weighted mean has a normal distribution. This paper compares

confidence intervals of Mw concentrations of ions in precipitation,

as computed using the assumption of a normal distribution, with those

estimated from distributions produced by bootstrapping. The

hypothesis that Mw was normally distributed was rejected about half

the time (at the 5{\%} significance level) in tests involving nine

major ions measured at ten diverse sites in the National Atmospheric

Deposition Program/National Trends Network (NADP/NTN). Most of these

rejections occurred at sites with fewer than 100 samples, in

agreement with previous results. Nevertheless, the hypothesis was

often rejected at sites with more than 100 samples as well. The

maximum error (relative to Mw) in the 95{\%} confidence limits made

by assuming a normal distribution of the Mw at the ten sites examined

was about 27{\%}. Most such errors were less than 10{\%}, and errors

were smaller at sampling sites with > 100 samples than at those with

< 100 samples.}}

Cheers,

JiHO

---

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